Models of Hyperbolic Geometry

Hyperbolic geometry was developed synthetically by Bolyai and Lobachevsky as a consistent-looking list of theorems — but looking consistent is not being consistent. A model settles the matter: it interprets the primitive terms ("point," "line," "congruent") as concrete objects inside ordinary Euclidean / real-number geometry, in such a way that all the hyperbolic axioms come out true. Any contradiction derivable in hyperbolic geometry would then be a contradiction in the host geometry — establishing relative consistency. This page presents the four standard models, the metric on each, and the dictionaries between them. All four are the same abstract hyperbolic plane ${\mathbb{H}}^2$ dressed in different coordinates.

References: Anderson, Hyperbolic Geometry; Beardon, The Geometry of Discrete Groups; Cannon–Floyd–Kenyon–Parry, Hyperbolic Geometry (Flavors of Geometry); Thurston, Three-Dimensional Geometry and Topology, ch. 2.

1. What a Model Must Do

To model hyperbolic geometry we must specify:

1. a set of points;
2. which subsets count as lines (geodesics);
3. how to measure distance and angle (a metric),

and then check the neutral axioms plus the hyperbolic parallel axiom hold. Two models are isometric if there is a bijection between them preserving lines and distance; all four below are isometric, so they describe one geometry. They differ in which features they render simple: some make geodesics look straight, others make angles look correct (conformality). We set the curvature radius $R=1$ ($K=-1$) throughout.

2. The Beltrami–Klein (Projective) Model

• Points: the interior of the open unit disk $\mathbb{D}=\{x^2+y^2<1\}$.
• Lines: open chords — straight Euclidean line segments with endpoints on the boundary circle (the boundary itself is not included; its points are the ideal points at infinity).
• Metric: the Cayley–Klein distance, expressed via the cross-ratio (see §6).

Virtue: geodesics are literally straight, so parallels are easy to see — given a chord $\ell$ and a point $P$ off it, the whole fan of chords through $P$ missing $\ell$ is visibly infinite. This makes the failure of the parallel postulate graphic, and it is why this was Beltrami's and Klein's model of choice.

Vice: the model is not conformal — angles are distorted, so it is awkward for anything involving angle measurement. It descends directly from projective geometry via the Cayley–Klein construction.

3. The Poincaré Disk Model

• Points: the interior of the unit disk $\mathbb{D}$.
• Lines: arcs of circles orthogonal to the boundary circle (plus diameters, the degenerate case).
• Metric: $$d{s}^2=\frac{4(d{x}^2+d{y}^2)}{(1-x^2-y^2{)}^2}.$$

Virtue: the model is conformal — hyperbolic angles equal the Euclidean angles you see in the picture. This makes it ideal for depicting tilings (Escher's Circle Limit prints live here) and for complex-analytic arguments. The metric blows up as $x^2+y^2\to 1$: the boundary is infinitely far away, so the disk's edge is the circle at infinity.

The distance from the centre to a point at Euclidean radius $r$ is $d=2\mathrm{artanh}(r)=\ln \frac{1+r}{1-r}$, growing without bound as $r\to 1$ — confirming the disk is metrically infinite.

4. The Poincaré Upper Half-Plane Model

• Points: the upper half-plane $\mathbb{H}=\{(x,y):y>0\}$, or in complex notation $\{z\in \mathbb{C}:\mathrm{Im}z>0\}$.
• Lines: vertical rays and semicircles centred on the real axis (both meeting the boundary $\mathbb{R}\cup \{\infty \}$ orthogonally).
• Metric: $$d{s}^2=\frac{d{x}^2+d{y}^2}{y^2}.$$

Virtue: also conformal, and its isometry group is beautifully explicit. Orientation-preserving isometries are the Möbius transformations with real coefficients,

$$z\mapsto \frac{az+b}{cz+d},a,b,c,d\in \mathbb{R},ad-bc=1,$$

i.e. the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I\}$. This is the single most useful model for group theory, number theory (modular forms), and the connection to special relativity; it is developed further in isometry-groups.md. The area element is $dA=\frac{dxdy}{y^2}$, giving the Gauss–Bonnet defect–area law directly.

5. The Hyperboloid (Minkowski) Model

• Points: the upper sheet of the two-sheeted hyperboloid in Minkowski space ${\mathbb{R}}^{2,1}$, $$H=\{(t,x,y):t^2-x^2-y^2=1,t>0\},$$ with the Minkowski form $⟨u,v⟩=u_0{v}_0-u_1{v}_1-u_2{v}_2$.
• Lines: intersections of $H$ with planes through the origin (the analogue of great circles for the sphere).
• Metric: the restriction of the Minkowski form to $H$ is positive-definite and gives constant curvature $K=-1$; distance is $\cosh d(u,v)=⟨u,v⟩$.

Virtue: the isometry group is manifestly the Lorentz group $O(2,1)$ (the linear maps preserving the Minkowski form and the upper sheet), making this the model that ties hyperbolic geometry to physics. The upper hyperboloid $t^2-x^2-y^2=1$ is exactly the mass shell / four-velocity space of special relativity: relativistic velocity space is a hyperbolic space, boosts are hyperbolic translations, and rapidity is hyperbolic arc length. The Beltrami–Klein disk is the central projection of this hyperboloid onto the plane $t=1$.

6. Dictionaries Between the Models

All four are isometric; the standard maps:

• Hyperboloid $\to$ Klein: central projection from the origin onto $t=1$ sends geodesics (plane sections) to chords — explaining why Klein lines are straight.
• Hyperboloid $\to$ Poincaré disk: stereographic projection from the point $(-1,0,0)$ onto $t=0$; stereographic projection is conformal, explaining why the Poincaré disk preserves angles.
• Disk $\leftrightarrow$ half-plane: the Möbius map $z\mapsto \frac{z-i}{z+i}$ (and its inverse $w\mapsto i\frac{1+w}{1-w}$) carries the upper half-plane conformally onto the unit disk, matching their metrics.

The Klein distance is given by the projective cross-ratio of the two points with the two ideal endpoints $A,B$ of their chord:

$$d(P,Q)=\frac{1}{2}|\ln (A,B;P,Q)|,$$

which is the Cayley–Klein metric — the thread linking these models back to projective geometry.

7. The Pseudosphere and Its Limitation

Beltrami's original 1868 realization used the pseudosphere, the surface of revolution of a tractrix, which has constant curvature $K=-1$ and so is locally isometric to the hyperbolic plane. It gives a tangible, sitting-in-space picture of negative curvature. Its limitation is essential:

Hilbert's theorem (1901). There is no complete, $C^2$-regular surface of constant negative curvature embedded in Euclidean ${\mathbb{R}}^3$.

So the pseudosphere models only a piece of ${\mathbb{H}}^2$ (it is incomplete, with a singular edge). The whole hyperbolic plane cannot be embedded isometrically in ${\mathbb{R}}^3$ — which is exactly why the abstract disk/half-plane/hyperboloid models are indispensable. Hilbert's obstruction is a first hint that intrinsic geometry (curvature.md) outruns what any surface-in-space picture can show.

8. What the Models Establish

Each model realizes hyperbolic geometry using only points, curves, and a metric built from Euclidean / real-number data. Therefore any contradiction inside hyperbolic geometry would produce one inside real analysis. Since we accept the real numbers as consistent, hyperbolic geometry is consistent too — and, as the next page draws out, this proves the parallel postulate independent of the remaining axioms, closing the two-thousand-year problem.